---

Source and scope. Part III, printed page 201 (PDF page 5), of Erdős (1974), read in the PDF's text layer; the quoted sentences were checked against the page image. This page quotes the passage's definition and its two questions and records the reported bounds in the corpus's words. It reconstructs no proof: the bounds are reported there from the Erdős–Hall paper the source cites as its [1] and from an unreferenced recent result of Hall, and neither paper is held or checked here.

Definition. For real $x$, Erdős writes $f(x)$ for "the number of integers $m<x$ for which $\varphi(n)=m$ is solvable" (p. 201), the count of distinct totient values below $x$. The catalog's $V(x)$ counts the values $n\le x$ with the letters swapped, so $V(x)-f(x)\in\{0,1\}$, the difference being the indicator that $x$ is itself an integer totient value; every asymptotic statement below is insensitive to it.

Reported bounds. Erdős reports that he and Hall proved, for every $k$ and every $\epsilon>0$, display (1):

$$\frac{x}{\log x}(\log\log x)^k<f(x) <\frac{x}{\log x}\,e^{(\log\log x)^{1/2+\epsilon}},$$

with the remark that the upper bound in (1) is probably nearly sharp, though no proof of that is in sight, and that Hall then proved display (2), $f(x)>x(\log\log x)^{c\log\log\log x}/\log x$.

Questions. Two sentences follow the bounds (p. 201): "It is not immediately clear if there is an asymptotic formula for $f(x)$ in terms of elementary functions. I can not prove that $\lim_{x=\infty}f(2x)/f(x)$ exists; if it exists it must be 2." These are the two questions of #416, whose site page cites this paper as [Er74b]; the doubling question is the one answered by the 2026 Lean proof recorded on that page. The page's next display, (3), asks whether $\lim A(x)/f(x)$ exists for the number $A(x)$ of totient values $n<x$ all of whose preimages exceed $x$; that reads as the complement form of the ratio $V'(x)/V(x)$ of #417 and is not extracted here.

Dependencies. None; a record of statements, not a deduction.

Bears on. #416.